In this circumstance it seems that an exclamation point would be the appropriate punctuation.
Regarding whether the sentences mean the same thing, I would say yes they generally do, but they come off slightly differently to me.
When I hear "God, I miss Europe!" I would tend to associate that with you missing the general place as a whole. That is, the culture, food, people etc. If you said this to me, I would expect our following conversation to be about Europe as a whole. What countries have the best food, beer, etc.
Conversely, "God, I miss being in Europe!" comes off a bit more personally to me. It sounds as if you miss the specific experiences you were having there. Were you to say "God, I miss being in Europe!" to me, I would expect the follow up conversation to be about things you did while you were there. Specific places you visited, specific meals you ate, people you met, etc.